If x1 and x2 are irrational numbers, can x1^x2 be a rational?

Question

Write 0 and then concatenate the prime numbers in increasing sequence i.e. 0.2357111317192329...
Is this number rational or irrational?

Note: Give your reasoning also...

Answer 1

We know that e is a transcendental and irrational number.
Also natural logarithm of any rational number is irrational.
This means e and ln(any rational number) are both irrational
Then e^(ln (2)) = 2 which is rational.
So it's possible to have a rational answer for irrational^irrational.